Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ \theta=\pi / 4 $$

Step-by-Step Solution

Answer

The Cartesian equation is \(y = x\). Its graph is a 45-degree line through the origin.

Step 1: Convert Polar Equation to Cartesian Coordinates

The given polar equation is \(\theta = \frac{\pi}{4}\). In polar coordinates, \(\theta\) represents the angle formed by the line joining the origin to the point \((r, \theta)\). To convert this to Cartesian coordinates, recall that \(x = r \cos \theta\) and \(y = r \sin \theta\). However, \(\theta\) in this case is constant, so we'll use the trigonometric identity \(\tan \theta = \frac{y}{x}\). Thus, \(\theta = \frac{\pi}{4}\) implies \(\tan \frac{\pi}{4} = \frac{y}{x} = 1\). This simplifies to the equation \(y = x\) in Cartesian coordinates.

Step 2: Sketch the Graph of the Equation

The equation \(y = x\) represents a straight line that passes through the origin with a slope of 1. In the Cartesian coordinate system, this is simply a 45-degree line through the origin. In the polar coordinates \(r\theta\)-plane, since \(\theta = \frac{\pi}{4}\) is a constant, the line radiates outwards from the origin at an angle of \(\pi/4\), forming the same visual pattern as the Cartesian line \(y = x\), but extending indefinitely in both directions.

Key Concepts

Polar Coordinates

Polar coordinates are a way to describe a point's location in a plane using its distance from the origin and the angle from the positive x-axis. Instead of using a grid of x and y values (as in Cartesian coordinates), polar coordinates use:

• The radius ( r ): the distance from the origin to the point.
• The angle ( θ ): the angle made with the positive x-axis.

The point is then represented as ( r, θ ). This system is particularly useful in scenarios involving circular or rotational symmetry, as it naturally aligns with the properties of angles and curves.
In the problem you're exploring, θ = π/4 signifies that any point defined by this equation forms a 45-degree angle with the x-axis.

Cartesian Coordinates

Cartesian coordinates describe a point's location in a plane with two perpendicular axes, traditionally labeled as x and y. Each point is identified by its horizontal (x) and vertical (y) distance from the origin. This system is intuitive when dealing with rectangular shapes, straight lines, or linear graphs.
Converting from polar to Cartesian involves using relationships between trigonometry and geometry:

• x = r cos(θ)
• y = r sin(θ)

In your original problem, the given polar equation θ = π/4 was transformed directly using the identity an(θ) = y/x resulting in y = x , a straight line in Cartesian coordinates.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to each other. They are essential for converting between coordinate systems, simplifying expressions, and solving equations.
The key identity used in the problem is:

• an(θ) = y/x

For an angle θ, tangent represents the ratio between the opposite side (y) and the adjacent side (x) of a right triangle. In the case of θ = π/4, an(π/4) = 1 since both sides are equal (y = x). This identity was crucial in translating the polar equation into a familiar linear relationship in Cartesian terms.

Graph Sketching

Graph sketching involves plotting points or lines expressed by equations to visualize their relationships. It's a fundamental skill in understanding mathematical concepts, providing insights into the behavior of functions and their intersections.
For the exercise in question, the graph of the polar equation θ = π/4 was drawn as a line y = x in Cartesian form. This line passes through the origin with a 45-degree slope, illustrating symmetry across the origin.
In the context of the rθ -plane, θ = π/4 results in a line that emanates from the origin at a consistent angle, demonstrating the seamless transition between polar and Cartesian representations.